On the unsolvability of certain equations of Erdős-Moser type (Q2418876)

\textit{B. C. Kellner} [J. Number Theory 131, No. 6, 1054--1061 (2011; Zbl 1267.11031)] conjectured that for \(m>3\) the ratio \(S_k(m+1)/S_k(m),\) where \(S_k(m)=1^k+2^k+\dots+(m-1)^k,\) is never an integer. Since \(S_k(m+1)=S_k(m)+m^k\) an alternative formulation is that for any positive integer \(a\) the Diophantine equation \(aS_k(m)=m^k\) has no solution \((m,k)\) with \(m>3.\) In case \(a=1\) this reduces to the well-known Erd\H{os}-Moser equation (for a survey, see, e.g., [\textit{P. Moree}, Rocky Mt. J. Math. 43, No. 5, 1707--1737 (2013; Zbl 1362.11045)]). In this well-written article the author considers the equation \(aT_k(m)=(2m+1)^k,\) with \(T_k(m)=1^k+2^k+\dots+(2m-1)^k,\) and cleverly exploits the close connection between these two Diophantine equations. In particular, she shows how so-called helpful pairs found in the past for the Kellner equation can be used to make progress on the odd power sum analogue. This allows her to obtain very direct analogues of the results earlier obtained on the Kellner equation by her and the reviewer [\textit{I. N. Baoulina} and \textit{P. Moree}, in: From arithmetic to zeta-functions. Number theory in memory of Wolfgang Schwarz. Cham: Springer. 1--30 (2016; Zbl 1407.11048)]. Her main theorem states that if \(a\) is even or \(a\) has a regular prime divisor or \(2\le a\le 1500,\) then the equation \(aT_k(m)=(2m+1)^k\) has no solution \((m,k)\) with \(m>1.\) The case \(a=1\) remains open (the same applies for the Erd\H{os}-Moser equation), but it is shown that if a non-trivial solution exists, it must be very large and have quite special properties.